3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Abstract

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-$3$-isogeny on $A$ factors as a composition of $3$-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than $1$. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension $1$, we deduce that if $E/F$ is an elliptic curve admitting a $3$-isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank $0$ and a positive proportion have $3$-Selmer rank $1$. These results on bounded average ranks in families of quadratic twists represent new progress toward Goldfeld’s conjecture—which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$—and the first progress toward the analogous conjecture over number fields other than $\mathbb{Q}$. Our results follow from a computation of the average size of the $\varphi$-Selmer group in the family of quadratic twists of an abelian variety admitting a $3$-isogeny $\varphi$.